Determining azimuth of a transponder by measuring a plurality of phase shifts

ABSTRACT

A method for determining the azimuth α of a transponder relative to a radar system which transmits a frequency F varying linearly with time. The system comprises two receiving antennas disposed at a distance d from each other and produces two beat signals Fb 1  and Fb 2 , formed by mixing the transmitted wave with each of the two echo waves received from the transponder. The phase shift φ between Fb 1  and Fb 2  at a first time t, and frequency f, and the phase shift φ&#39; o  at a second time t 2  and frequency f 2  is measured, and the overall phase shift φ between Fb 1  and Fb 2  is calculated. The value of α is calculated from the values F 1 , d and φ.

BACKGROUND OF THE INVENTION

The invention relates to a method of determining the azimuth α of aradiowave transponder relative to a radar system which transmits a wavehaving a frequency F which varies linearly with time. The radar systemincludes a transmitting antenna and two receiving antennas, which arelocated at a distance d from each other. Two beat signals Fb₁ and Fb₂having frequencies fb₁ and fb₂, respectively, are obtained by mixing thetransmitted wave and the echo wave received from the transponder at eachof the two receiving antennas.

The invention also relates to an apparatus for measuring the azimuth αusing the above method. The apparatus forms part of a radar system whichtransmits a high frequency continuous wave, which is frequency modulatedwith a sawtooth having a constant frequency sweep ΔF and a duration T,and which simultaneously receives the previously transmitted wave whichis returned by a transponder. The radar system supplies a signal Fb₁ ofa first beat frequency fb₁, obtained by mixing the transmitted signal ofthe instantaneous frequency F and the signal received at a firstreceiving antenna, and a signal Fb₂ of a second beat frequency fb₂,obtained by mixing the transmitted signal of the frequency F and asignal received at a second receiving antenna. The reference directionfor measuring the azimuth α of the transponder is perpendicular to aline section of length d at whose ends the receiving antennas arelocated.

The azimuth angle α to be determined is the angle between apredetermined direction, for example a reference axis associated withthe apparatus for measuring α, and an axis which extends from themeasuring apparatus to a target whose angular location is to bedetermined. Suitably, the measuring station is located on the ground,the measuring apparatus comprises a radar interrogator, and the targetis an aircraft equipped with a transponder. The measuring station mayalternatively be an aircraft. In practice, the angle α to be determinedis suitably the angle between the mid-perpendicular plane to thereceiving antennas of said radar system and the axis between the radarsystem and the target. On the other hand, the transponder associatedwith the target may be a simple passive reflector, in so far that it isisolated in the space surrounding it.

The apparatus used for carrying out the invention may, as far as theradar interrogator is concerned, for example be of the type known fromFrench Patent Specification No. 1,557,670 corresponding to U.S. Pat. No.3,588,899. The radar system comprises a second receiving antenna bymeans of which a second beat signal Fb₂ of the frequency fb₂ is obtainedby mixing the transmitted wave and the wave received by the secondreceiving antenna in a second mixer. Such a radar system serves as adistance measuring apparatus and to this end it comprises a control loopwhich maintains the first beat signal Fb₁ at a substantially constantfrequency fb₁ as the distance varies. This results in a variation of theduration of the transmitted sawtooth as a linear function of thedistance for a constant frequency sweep Δf of the sawtooth.

It is to be noted that the invention is not limited to this type ofapparatus. It equally applies to a radar system which transmits asawtooth of constant frequency, duration and frequency sweep and whichsupplies two beat signals Fb₁ and Fb₂ obtained by mixing of thetransmitted wave and the echo wave received from the transponder.

The transponder used is for example of the type described in FrenchPatent Specification No. 2,343,258, corresponding to U.S. Pat. No.4,151,525 in particular with reference to FIGS. 9 and 10, by means ofwhich the azimuth of the target can be calculated at distances greaterthan 100 km.

By means of the two distance measuring apparatus of the type describedin the French Patent Specification No. 1,557,670, having a commontransmitting antenna and each having one receiving antenna, the azimuthcan be determined in known manner from two distances measured bytriangulation using the formula: ##EQU1## in which:

d is the (fixed) distance between the receiving antennas

R₁ is the distance between the transponder and one receiving antenna

R₂ is the distance between the transponder and the other receivingantenna.

The principle of determining α is described in more detail in thepreviously mentioned French Patent Specification No. 2,343,258.

When α is thus determined this has the drawback that at least onedistance measuring apparatus is necessary (by alternately switching thecontrol loop from one receiving antenna to the other in which case thefrequencies fb₁ and fb₂ are equal) and that the measurement of α is notvery accurate because of the length of the signal-processing chainnecessary to enable the distances R₁ and R₂ and their difference to bedetermined, which leads to an accumulation of the absolute errorsproduced by the various signal-processing elements, the cumulative errorincreasing as the distance R increases.

It is also possible to determine the angle α by means of the formula:##EQU2## c being the velocity of propagation of an electromagnetic wave.

Such a method of determining α by measuring T, fb₁ and fb₂ has the samedrawbacks as described in the foregoing.

SUMMARY OF THE INVENTION

It is an object of the invention to obtain a comparatively high accuracymeasurement of α, for example of some hundredths of degrees, usingsimple radar equipment. The accuracy should be of the same order ofmagnitude as that obtained by means of an aircraft landing radar system(ILS system). More specifically, it is an object of the invention toobtain this high accuracy by means of only two antennas, while inconventional angle measuring systems this is achieved by means of alarge number of antennas (interferometers with a plurality of antennas).

In accordance with the invention, the method defined in the openingparagraph comprises the following steps:

The algebraic measurement of the phase shift φ_(o) between the signalsFb₁ and Fb₂ and the measurement of the frequency F₁ at a predetermined,arbitrarily chosen first instant t₁,

The algebraic measurement of the phase shift φ'_(o) between the signalsFb₁ and Fb₂ and the measurement of the frequency F₂ at a secondarbitrarily chosen instant t₂, φ'_(o) -φ_(o) being such that the numberof sinewave periods of fb₁ and fb₂ between the two instants t₁ and t₂ issubstantially the same,

The calculation of a relative overall phase variation Δφ which may begreater than 2π between the instants t₁ and t₂, by determining thedifference between φ'_(o) and φ_(o),

The approximated calculation of φ, which is the overall phase shiftbetween the signals Fb₁ and Fb₂ at the first instant t₁, the time originbeing when the frequency F is zero, as a function of F₁, F₂ and thevalue of Δφ found in the preceding step, that is φ.sub.Δφ,

The determination of the maximum angle 2kπ, k being a positive integerwhich is actually contained in the angle φ, from φ_(o), φ.sub.Δφ foundin the preceding step and of the respective signs of φ_(o) and of Δφ,

Making φ identical to the sum: φ_(o) +2kπ or φ_(o) -2kπ depending on therespective signs of φ₀ and of Δφ,

The calculation of sin α from the values of F₁, d and the exact value ofφ found in the preceding step,

The calculation of α from the value of sin α obtained in the precedingstep,

The display of the value of α found in the preceding step.

Similarly, in order to obtain a high accuracy for α, the apparatusdefined in the introduction is characterized in that it comprises:

First means for shaping the signals Fb₁, Fb₂ of the frequencies fb₁ andfb₂ to obtain squarewave signals of the same phase and the samefrequency,

Second means for measuring the phase shift φ₀ between the squarewavesignals of the frequencies fb₁ and fb₂, as well as the frequency F forat least one point of said sawtooth,

Third means for determining at least two trains of squarewave signalshaving the same number of periods and whose starting points differ byless than one period,

Fourth means for measuring the overall relative phase variation Δφbetween the beginning, at the instant t₁ for a frequency F₁, and theend, at the instant t₂ for a frequency F₂, of the trains of sqaurewavesignals,

Fifth means for calculating the angle α from the values of F₁, F₂, d,φ_(o) and Δφ and for displaying the angle.

By means of a simple formula, mentioned in the detailed description, itis possible to calculate the value of sin α for a given point of thesawtooth from the value φ of the overall phase shift between Fb₁ and Fb₂with the required accuracy. The value φ_(o) measured for this point ofthe sawtooth only represents the portion of φ which is smaller than 2π.

The basic concept of the invention is to determine the angle 2kπ which,when added to φ_(o), yields the angle φ. This is possible by measuringthe phase shift φ'_(o) for at least a second point of the sawtooth.Thus, if the phase shifts φ_(o), φ'_(o) are measured with an accuracy ofthe order of, for example, 1°, that is a relative error of the order of0.5%, it is possible to obtain the angle φ with a much higher relativeaccuracy. Indeed, the overall phase shift between Fb₁ and Fb₂ isobtained with a relative accuracy on the order of 1° in severalthousands of degrees. This high accuracy is then also obtained for α.

BRIEF DESCRIPTION OF THE DRAWINGS

The following description, with reference to the accompanying exemplarydrawings, enables the invention to be more fully understood.Corresponding elements bear the same reference numerals.

FIG. 1 is the simplified block diagram of a radar system whichsimultaneously transmits and receives a high frequency continuous wave,which is frequency-modulated as a sawtooth and which provides thesignals necessary for carrying out the invention.

FIG. 2 represents the frequency variation of the transmitted andreceived signals as a function of time.

FIG. 3 is the block diagram of an embodiment of the invention.

FIG. 4 is a time diagram illustrating the operation of the circuitsshown in FIGS. 1 and 3.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 represents a radar system 1, which may be a radio altimeter or adistance measuring apparatus using high-frequency continuous waves whichare frequency-modulated in accordance with a sawtooth waveform. Thesystem comprises a transmitting antenna 2 as well as two receivingantennas 3 and 4 situated at a distance d from each other. The radarsystem 1 forms part of a system which further comprises a transponder 5,represented at the left in FIG. 1, whose distance 6 from the radarsystem may exceed 140 km. The transponder 5 suitably comprises a singletransmitting/receiving antenna 7 at o'. In order to ensure that the wavereceived from the antenna 2 is returned to the antennas 3 and 4 of theradar system 1 with sufficient power, especially in the case of longerdistances, the transponder 5 is suitably of the type described in FrenchPatent Specification No. 2,343,258, in particular with reference toFIGS. 9 and 10 of said Specification, or a transponder of comparabledesign and performance. This type of transponder comprises a delay linewhich provides a delay τ_(o) of microsecond order between the receivedsignal and the retransmitted signal, an amplifier, and means, in theform of at least one radio-frequency switch for sampling the receivedsignal at a frequency of the order of some hundreds of kilohertz. Theradar system 1 is adapted to analyze the signals returned to itsreceiving antennas 3 and 4 by the transponder 5 in order to obtainoutput signals which, in accordance with the invention, enable the valueof the angle α, which is the azimuth angle of the transponder relativeto the radar system, to be determined with an accuracy of the order ofsome tens of degrees. In FIG. 1, α is the angle between a direction oz,perpendicular to a line section of length d which interconnects thecenters of the antennas 3 and 4, and the direction oo.

The transmitting section of the radar system 1 comprises avoltage-controlled oscillator 8 connected to the transmitting antenna 2.The input of the oscillator receives the output signal of a sawtoothgenerator 9.

The receiving section is constituted by two identical signal processingchains. The first chain comprises a mixing circuit 10 having a firstinput connected to the output of the antenna 3 and a second inputconnected to the output of the oscillator 8 via a coupler 11. An outputof the mixing circuit 10 is connected to cascaded elements including aband-pass filter 12, an amplifier 13 and an amplitude limiter 14. Themixer 10 forms the difference frequency of the transmitted wave and thereceived wave, yielding a signal Fb₁ of frequency fb₁ on an outputterminal 15, which frequency is the instantaneous difference frequencybetween the wave transmitted at 2 and the wave received at 3. Like thefirst chain, the second chain includes cascaded elements including thereceiving antenna 4, a mixer 16, a band-pass filter 17, an amplifier 18and an amplitude limiter 19. The second input of the mixer 16 is alsoconnected to the coupler 11. The output of the amplitude limiter 19produces a signal Fb₂ of frequency fb₂ on its output terminal 20, whichfrequency is the instantaneous difference between the frequencies of thewave transmitted at 2 and the wave received at 4. The radar system 1also comprises two elements 21 and 22 which receive via a conductor 28the output voltage of the sawtooth generator 9. The element 21 is alogic signal generator which produces signals DE and S on outputs 23 and24, and the element 22 produces a signal F (λ) on a terminal 25. Thesignals DE and S are also supplied to the element 22. The function ofthese elements 21 and 22 will be described hereinafter with reference toFIGS. 3 and 4. FIG. 1 also shows a frequency discriminator 26, operatingat a central frequency f_(o), and an integrator 27 arranged in cascadebetween the output of the amplitude limiter 14 and a control input ofthe sawtooth generator 9. Their presence is optional, which is indicatedby the broken lines, and their function will be described hereinafter.

FIG. 2 represents frequency curves as a function of time. The curve EMrepresents the signals transmitted at 2 and 11 and the curves RE₁ andRE₂ represent the signals received at 3 and 4.

The curve EM has the form of a sawtooth with a fixed or variableduration T and a frequency sweep ΔF, which is preferably constant.Although an asymmetric sawtooth is shown, a symmetric sawtooth may alsobe used. The frequency F_(o) is the frequency at the beginning of thesawtooth. In practice F_(o) is of the order of magnitude of GHz and ΔFis of the order of magnitude of ten or some tens of MHz. When theDoppler effect is ignored and the waves received at 3 and 4 arecontinuous, the curves RE₁ and RE₂ can be derived from the curve EM by atranslation parallel to the time axis, with a duration τ₁ and τ₂,respectively. Referring now to FIG. 1, τ₁ is the time which the waveneeds to cover the distance R between the antennas 2 and 7, to passthrough the transponder 5 (time τ_(o)) and to cover the distance R₁between the antennas 7 and 3 in the other direction, namely: ##EQU3##Similarly: ##EQU4## The respective beat frequencies fb₁ resulting frommixing RE₁ and EM, and fb₂ resulting from mixing RE₂ and EM may berepresented by the formulas: ##EQU5## It is to be noted that RE₁ and RE₂are only the envelopes of the waves received by the radar system.Indeed, for the given type of transponder which is preferably used, thewave returned by the latter is chopped at the sampling frequency, thatis for each sampling cycle of a duration which is typically 2 μs, it isonly present on the output of the transponder 5 for approximately 1 μs.It follows that the beat signal of the frequency fb₁ (or fb₂) at theoutput of the mixer 10 or 16 itself is sampled at the sampling frequencyof the transponder, which is of the order of 500 kHz. The function ofthe band-pass filter 12 or 17 is to recover the beat signal in the formof a sinewave of the frequency fb₁ or fb₂ by eliminating the componentsof the sampling frequency and multiples thereof from the spectrum of thesignal which is received. This is possible if the frequencies fb₁ andfb₂ are smaller than half the sampling frequency, that is, for example250 kHz (Shannon theorem).

When the sawtooth is constant (T and ΔF constant), the criterion givenin the preceding paragraph imposes a limitation on the distance betweenthe radar system and the transponder in view of formulas (3) and (4). Inorder to avoid this limitation, the sampling frequency may be increased(by reducing the duration τ_(o) of the transponder) and/or the ratio(ΔF/T) may be reduced by influencing the values of ΔF and T in the radarsystem, in such a way that the distance limit imposed by the samplingfrequency becomes greater than the distance limit imposed by the maximumgain of the transponder 5.

In a preferred embodiment of the invention one of the beat frequenciesfb₁ or fb₂ is maintained substantially equal to a constant frequencyf_(o) by means of a control loop in the transmitting section of theradar system. This control loop includes the oscillator 8, the coupler11, the mixer 10, the filter 12, the amplifier 13, the amplitude limiter14, the frequency discriminator 26, the integrator 27, and the sawtoothgenerator 9. The output signal of the discriminator 26 influences thegenerator 9 via the integrator 27 in such a way that the slope of thesawtooth varies as a function of the distance to the transponder, tomaintain the frequency fb₁ constant. In this type of distance measuringapparatus, which is for example known from the French PatentSpecification No. 1,557,670, the duration T of the sawtooth is a linearfunction of the distance between the transponder and the radar system,thereby enabling this distance to be measured. The control loopfunctions to ensure that substantially constant values are obtained forfb₁ and fb₂ (the value of fb₂ being very close to that of fb₁)independently of the distance between the radar system and thetransponder, which ensures that the sampling theorem is complied with.In practice, the frequency f_(o) is of the order of some tens ofkilohertz, that is, an order of magnitude smaller than the samplingfrequency.

Referring now to the left-hand part of FIG. 1, the line sectionsrepresenting the distances R, R₁ and R₂ from the antenna 7 to theantennas 2, 3 and 4, respectively, are sufficiently long that they maybe considered to be parallel to a reasonable approximation. As a resultof this, the line perpendicular to the line 00' (and to the linesections R₁ and R₂) from the center of the antenna 3 makes an angle αwith the line section of the length d. It follow that: ##EQU6## On theother hand, subtracting formulas (3) and (4) from each other yields:##EQU7## The difference fb₁ -fb₂, may be expressed as a number ofperiods which linearly increase with time or rather an "overall phaseshift" φ, whose absolute value is greater than 2π, which may beexpressed by:

    φ=2π(fb.sub.1 -fb.sub.2)t                           (7)

when taking a suitable origin for t, that is for each sawtooth thepoints o" where the line corresponding to the curve EM in FIG. 2intersects the horizontal axis.

Formula (7) may be written as follows using formula (6): ##EQU8## Theexpression for the curve F as a function of time for each sawtooth isthen: F=(ΔF/T) t when taking the same origin o" as above for t. Formula(8) may then be written as follows: ##EQU9##

Combining equations (5) and (9). yields: ##EQU10## In equation (11) thevalues of F (or of λ=(c/F)) and d are known with excellent accuracy, butthe angle φ cannot be measured directly: it is only possible to measureits algebraic value φ_(o) with a suitable accuracy of the order of onedegree, whose period is smaller than 2π and whose sign is either thatfor φ (and thus for α) or the opposite sign. Measuring φ_(o), which isactually a phase measurement, therefore gives rise to an indeterminatefactor and does not suffice for a correct evaluation of φ with anaccuracy of one degree, because the absolute value of the angle φ is ofthe order of some hundreds to some thousands of degrees.

The angle φ may therefore be expressed as a function of φ_(o) by meansof one of the following two formulas:

    φ=φ.sub.o +2kπ if φ is positive             (12)

    φ=φ.sub.o -2kπ if φ is negative

where k is a positive integer.

In order to remove the ambiguity associated with the measurement ofφ_(o), it is to be noted that because F varies during the sawtoothmodulation φ will also vary, so that for example between the beginning(φ₁, F₁) and the end (φ₂, F₂) of the sawtooth:

Δφ=φ₂ -φ₁, so that because of formula (10): ##EQU11## Δφ is anelectrical angle which has the sign of α and which for the use inaccordance with the invention rarely exceeds 2π. It is to be noted thatwhen Δφ is greater than 2π, its value can be measured because itconcerns the variation of the relative phase shifts of the two signalsduring a given interval of time which only comprises a fairly limitednumber of periods for the signals Fb₁ and Fb₂.

For an accuracy of the measurement of Δφ comparable to the accuracyobtained for φ_(o), that is approximately one degree, formula (13)yields an accuracy for sin α which is lower than formula (11), as willbe seen hereinafter, but on the other hand this enables sin α to bedetermined without ambiguity.

According to the invention the amplitude and sign of the angle Δφ ismeasured. This sign is also that of α and thus of φ because of formulas(14) and (11). The value of sin α is calculated from formula (14), whichvalue is designated sin α.sub.Δφ. The value of sin α.sub.Δφ is insertedin formula (10) and a first approximated value of the angle φ iscalculated therefrom, which is designated φ.sub.Δφ. On the other handφ_(o) is also measured and is preferably made identical to φ₁.Comparison of the signs of Δφ and φ_(o) makes it possible to decidewhich of the formulas (12) is valid for the determination of k (φ and Δφhave the same signs). For example, if the second of these formulas isvalid, the value of k is defined as the integer nearest the calculatedvalue, which is equal to ##EQU12## Now φ is calculated in an inversemanner by means of the same formula (12) with which k has beendetermined, using the integer found for k and, finally, thelast-mentioned correct value found for φ is inserted into formula (11),which then enables the value of sin α and thus the value of α to becalculated with the desired accuracy. Differentiation of formula (10)yields: ##EQU13## which, when assuming for example that: d=4 m and F=F₁=1.22 GHz, yields:

    for α=0  d=0.0097dφ

    for α=30° d=0.0112dφ

that is, an error of ±1° where φ corresponds approximately to 0.01° forα.

Conversely, if α is to be determined from the value of Δφ only,differentiation of formula (13) yields: ##EQU14## or when it is forexample assumed that: d=4 m and ΔF=10 MHz (F₁ =1.22 GHz, F₂ =1.23 GHz):##EQU15##

In this case the accuracy obtained for α varies for ±1.2° for α=0 to±1.4° for α=30°, with an accuracy of ±1° for Δφ. Thus, this is clearlyinsufficient in comparison with the desired accuracy.

It is to be noted that for d=4 m and F₁ =1.22 GHz, the angle φ varies by2π when α varies by 3.4 degrees at 0° or by 4 degrees at 30 degrees. Theaccuracy obtained for α by means of formula (16) is therefore sufficientto ensure that the correct value of k can be determined by means of oneof the formulas (12). If the accuracy is no longer sufficient, this maybe solved by increasing the value of d and/or that of ΔF.

An embodiment of the invention, which employs the measuring andcalculation method explained in the foregoing, is now described withreference to FIGS. 3 and 4. In this embodiment the phase shifts arepreferably measured by the comparison of counted numbers of clockpulses, the number of pulses being counted between the zero passages ofthe beat signals Fb₁ and Fb₂.

The instants marking the beginning and end of the phase measurementduring a sawtooth may be selected arbitrarily, provided that thewavelength or frequency emitted at these two instants is known. Thefirst instant is for example selected to correspond to the beginning ofthe sawtooth and the second instant to correspond to 90% of theexcursion of the sawtooth or: ΔF'=0.9ΔF.

The device of FIG. 3 comprises two identical signal-processing chains,whose inputs respectively receive the signal Fb₁ on terminal 15 and thesignal Fb₂ on terminal 20. The chain receiving a signal Fb₁ (Fb₂)comprises a cascade of: a shaping circuit 30 (40), which shapes thesinewave signal which it receives into squarewave signals, asynchronizing circuit 31 (41), an AND-gate circuit 32 (42), a periodcounter 33 (43), and a comparator 34. The outputs of the elements 30, 31and 32 supply the signals A₁, B₁, C₁, respectively. The signal A₁ (A₂)is supplied directly to a second input of the AND-gate circuit 32 (42).Furthermore, a first (second) output of the comparator 34 is connectedto an AND-gate circuit 35 (45), which at a second input receives thesignal S from terminal 24 and whose output is connected to a secondinput of the synchronizing circuit 31 (41). On a third input the circuit31 (41) receives the signal DE from the terminal 23. The signals B₁ andB₂ are applied to an exclusive-OR gate 50 and to a first switchingdetector circuit 51. The output of the gate circuit 50, on which thesignal E appears, is followed by a cascade of: an AND-gate 52, whichreceives the output signal of a fast clock generator 53 on the secondinput and whose output supplies a signal H, a pulse counter 54, a memory55, a computing element 56 for calculating Δ φsin α.sub.Δφ, φ.sub.Δφ, k,φ, sin α and α and a display element 57 for displaying the value of α.The circuit 51, whose function is to determine the signs of the measuredphase shifts, transfers said signs, for example in the form of logiclevels, to the memory 55 via two conductors. The computing element 56receives in digital form, the value of the distance d, which isdisplayed by an element 58, and the value of the transmitted frequency F(or the wavelength λ) which is transferred to terminal 25 at theinstants (t₁, t₂) which respectively correspond to the transition to thehigh level of the logic signals DE and S, which in FIG. 1 is indicatedby the conductors which connect each of the terminals 23 and 24 to acontrol input of the analog-to-digital converter 22.

The operation of the apparatus of FIG. 3 is described hereinafter withreference to FIG. 4, which is a time diagram of the signals EM, DE, S,Fb₁, A₁, B₁, C₁, Fb₂, A₂, B₂, C₂, E, H. In FIG. 4 the signal G is afixed frequency threshold, determined by the element 21 of FIG. 1, forexample equal to 90% of the peak value of the sawtooth (ΔF'=0.9 ΔF) andS is a logic signal which changes from 0 to 1 when the threshold G isreached and which returns to 0 at the end of the sawtooth.

The phase shift between the echo signals received by the antennas 3 and4 is imparted to the beat signals of the frequencies fb₁ and fb₂ bymeans of mixers 10 and 16 (FIG. 1). The best signals are available inthe form of continuous sinewaves at respective terminals 15 and 20 (FIG.1), in which form they are shown in FIG. 4. By means of the circuits 30,40 (FIG. 3) the signals Fb₁ and Fb₂ are shaped into squarewave signalsA₁, A₂ having amplitudes adapted to suit the following logic circuits(logic levels "0" and "1"). The synchronizing circuits 31, 41 haveoutputs B₁, B₂, which are 0 between the sawtooth waves and which becomes1 upon the first change from 0 to 1 of the signal A₁ or A₂ following theinstant t₁, beginning the sawtooth. For this purpose, the synchronizingcircuits 31, 41 receive the signal DE on its third or "start" input.When, at the instant t₂, each circuit 31, 41 receives a signal on itssecond or "stop" input, B₁ and B₂ will return to 0 upon the nexttransition from 0 to 1 of the respective signal A₁ or A₂. Such logiccircuits 31, 41 are known to those skilled in the art. The AND gates 32and 42, which receive the signals A₁, B₁ and A₂, B₂, produce integralnumbers N₁, N₂ of periods C₁, C₂ at their respective outputs. Thecounters 33, 43 each supply a number equal to the number of sinewaveperiods of Fb₁ or Fb₂ during the interval under consideration. When thesignal S changes to the 1 level causing the signals B₁ , B₂ to be resetto zero via the AND-circuits 35, 45, one of the two following modes ofoperation of the apparatus is possible.

(1) N₂ ≧N₁ (case considered in FIG. 4), in which case the comparator 34transfers a logic "1" via the AND-gate 45, which resets the output B₂ ofthe synchronizing circuit 41 to zero during the following passage from 0to 1 of the period A₂. The AND-gate circuit 35 remains closed(inhibited) until N₁ =N₂. At this instant the first output of thecomparator 34 also goes to 1, which pulls the synchronizing circuit 31to 0 via the gate circuit 35. The AND-gate circuits 42 and 32 thus havesupplied the same number of periods and the durations of the "1" levelsof the signals B₁ and B₂ represent the respective durations of the samenumber of sinewave periods in the two respective signal processingchains Fb₁ and Fb₂.

(2) N₁ ≧N₂, in which case the operations in the two chains described inthe foregoing are interchanged, and by means of the same reasoning thesame result is obtained as in the preceding paragraph.

The signals B₁ and B₂ are, for example, as shown in FIG. 4, but otherconfigurations are also possible because first either B₁ or B₂ switchesfrom the low level to the high level (first and second switchingoperation) and subsequently either B₁ or B₂ changes from the high levelto the low level (third and fourth switching operation).

The sign of φ₀ depends on the chronological sequence of the first andthe second switching operation. By convention, it is for example decidedto count φ₀ positively when the first switching operation takes place inthe first signal processing chain and negatively if this takes place inthe second chain. This convention, as will be seen hereinafter, allowsthe value of α to be determined in a trigonometric sense. In accordancewith this convention the angle φ₀ is negative in FIG. 4.

On the other hand, the difference between the durations of the highlevels of the signals B₁ and B₂ represents the absolute value of therelative overall phase shift Δφ. The absolute value and the sign of Δφcan be obtained by algebraically measuring φ'₀, that is, the algebraicdifference between the falling edges of the signals B₁ and B₂ (third andfourth switching operations) with the same sign convention as in theforegoing and by subtracting the algebraic value obtained for φ_(o) fromsaid algebraic value (first and second switching operations), which ruleis valid regardless of the configuration of the signals B₁ and B₂ . Thesign obtained for Δφ is also the sign of α because of formula (14).

In FIG. 4 the two measured phase shifts are negative, their difference(the second one minus the first one) is negative, which means that theangle α is negative when the axis oz is taken as the origin (which caseis represented in FIGS. 1 and 2). It is to be noted that FIGS. 1, 2 and4 represent the same case, for which the following inequality is valid:Fb₂ >Fb₁. If the beat frequency Fb₁ is maintained constant and equal toa predetermined value, for example 25 kHz (period of 40 μs), if B₁ has adelay of 10 μs at the beginning and of 20 μs at the end, then:

The initial phase shift φ₀ is: ##EQU16##

The final phase shift φ'₀ is: ##EQU17##

The variation of the phase shift Δφ is consequently: ##EQU18## When thevariation of the transmitted frequency, on terminal 25, between thebeginning and the end of counting is known, that is, F₂ -F₁, the valueof sin α can be calculated from this variation in a first approximation(accuracy of the order of one degree for arc sin α).

The actual circuit for measuring and calculating α from timemeasurements representing the phase shifts φ₀ and Δφ is shown in theright-hand part of FIG. 3 (the elements 50 to 58).

The exclusive-OR gate circuit 50 receives the two signals B₁ and B₂ andsupplies the signal E (FIG. 4), which for a given sawtooth comprises twopulses representing the initial and final phase shifts φ₀ and φ'₀. Viathe AND-gate circuit 52, which also receives the output signal of thefast clock generator 53, the signal E is converted into a counting pulsesignal H with a frequency of, for example, 20 MHz. At the end of eachtrain of pulses supplied by the AND-gate 52 the counter 54, which hasbeen reset to zero before the beginning of each pulse train by means notshown, provides the phase-shift value expressed by a number which is ameasure of the time which has elapsed between similar changes of thesignals B₁ and B₂.

The circuit 51 detects which circuit effects the first switchingoperation and, in accordance with the convention adopted, derivestherefrom a + or - sign, which is subsequently transferred in the formof logic signals.

At the end of the counting operation at 54 the number and sign arestored at 55, which is suitably a temporary-storage memory, for examplea buffer memory. The digital values are subsequently transferred to thecomputing element 56, which is suitably a microprocessor. As indicatedin the foregoing, the element 56 also receives, in digital form, thevalue of the frequency F or the wavelength of the transmitted signal aswell as the value of the distance d from the element 58. In achronological sequence the operations or calculations effected for eachsawtooth by 56 are the following:

making the first algebraic value from the memory 55 equal to φ₀ and thesecond algebraic value to φ'₀ ;

calculating Δφ by forming the difference between φ'₀ and φ₀ ;

calculating sin α from formula (14) (sin αΔφ);

calculating φ.sub.Δφ from formula (10);

selecting the formula (12) to be used as a function of the respectivesigns of φ₀ and Δφ;

making the approximated calculation of k from the appropriate formula(12) and determining k;

calculating φ from the same formula (12) using the integral value of k;

calculating sin α from formula (11);

calculating α as a function of sin α.

The value of α thus determined is transferred to the element 57, whichsuitably displays the value in digital form, for example in degrees andminutes or hundredths of degrees, with the aid of light emitting diodesor liquid crystals.

It is to be noted that the foregoing calculation of α may be simplifiedbecause, except for clarity of the explanation it is not necessary toinclude the approximated value of sin α, that is sin αΔφ. Combiningformulas (14) and (10) yields: ##EQU19## in which formula now only thevalues of F₁, F₂, φ₀ and φ'₀ occur, i.e. the actual measuring values.Regarding the accuracy which is obtained, it is to be noted that if theabsolute error is the same for Δφ and φ after accurate calculation ofthe latter, the ratio (ΔF/F) enables the relative error for Δφ to bemaintained for φ.sub.Δφ using the last-mentioned formulas, and that thehigh accuracy obtained for φ is maintained in the accuracy obtained forα.

In another embodiment of the invention, not shown, the accuracy obtainedfor the value of α may be further improved and reduced from a fewhundredths of a degree approximately to one hundredth of a degree incomparison with the example where an accuracy of one degree for thephase shift measurements is obtained. In this embodiment the phaseshifts of a plurality of pairs of sinewave periods of the signals Fb₁and Fb₂ are measured by each time taking the corresponding value of thefrequency (or wavelength) of the transmitted wave and assigning, to eachvalue of φ₀ thus obtained, the same angular value φ'₀, which isdetermined from the falling edges of the signals B₁ and B₂. Thus, bymeans of the computing element it is possible to determine for eachsawtooth as many values of sin α as the number of different valuesmeasured for the angle φ₀ and of the frequency F₁ corresponding thereto,each time taking the same value for φ'₀ and for F₂. In this case thecomputing element should perform an additional operation of a differentnature, which for determining α consists of previously determining themean of the different values found for sin α.

Suitably, the antennas 2, 3 and 4 shown in FIG. 1 are of the directionaltype and cover an angular sector on the order of 60 degrees. However,they may cover a larger angle such as 120 degrees, but when determiningthe angle α this may result in a smaller accuracy than in the case of acoverage of 60 degrees. The arrangement of the six devices as describedin the foregoing at 60 degrees from each other or of three devices at120 degrees from each other, depending on whether the angle of coverageof the antennas is for example 60 or 120 degrees, makes it possible tocover the entire plane.

What is claimed is:
 1. A method for determining the azimuth of a radiowave transponder relative to a radar system having first and secondreceiving antennas located at a distance d from each other, comprisingthe steps of:a. transmitting from the radar system to the transponder asignal having a frequency which varies linearly with time; b. returningfrom the transponder a signal representative of the transmitted signal;c. receiving the returned signal at the first and second receivingantennas at times which differ in relation to any differences in thedistances between said receiving antennas and the transponder; d. mixingthe signal received at each of the first and second receiving antennaswith the transmitted signal and producing first and second beat signalsFb₁ and Fb₂ having frequencies fb₁ and fb₂, respectively; e. during afirst interval following the transmission of a frequency F₁ at a timet₁, measuring the sign and magnitude of the phase shift φ₀ between thesignals Fb₁ and Fb₂ ; f. during a second interval following thetransmission of a frequency F₂ at a time t₂, measuring the sign andmagnitude of the phase shift φ'₀ between the signals Fb₁ and Fb₂, thetiming of said first and second intervals being chosen such that theintegral number of cycles of Fb₁ and Fb₂ between t₁ and t₂ aresubstantially equal; g. determining an overall phase variation Δφ byperforming the calculation: Δφ=φ'₀ -φ₀ ; h. determining an approximatevalue φ.sub.Δφ for the phase shift between the signals Fb₁ and Fb₂ atthe time t₁ by performing the calculation: ##EQU20## i. determining thevalue of a constant k to the nearest integer by performing theappropriate one of the following calculations:
 1. if the sign of Δφ ispositive, ##EQU21##
 2. if the sign of Δφ is negative, ##EQU22## j.determining the accurate value of the phase shift φ between the signalsFb₁ and Fb₂ by performing the appropriate one of the followingcalculations:
 1. if the sign of Δφ is positive, φ=φ₀ +2kπ2. if the signof Δφ is negative, φ=φ₀ -2kπ k. determining the azimuth α of thetransponder by performing the calculation: ##EQU23##
 2. A method as inclaim 1 where steps e-k are repeated for a plurality of successive timeintervals at times t_(i) occurring between t₁ and t₂ at a plurality ofrespective frequencies F_(i) effecting the calculation of a plurality ofazimuth values α_(i) and where the azimuth α is determined bycalculating the mean value of α_(i).
 3. An apparatus for measuring theazimuth of a radio wave transponder relative to a radar system,comprising:a. transmitting means for transmitting from the radar systemto the transponder a signal having a frequency which varies linearlywith time; b. means for returning from the transponder a signalrepresentative of the transmitted signal; c. first and second receivingantennas located at a distance d from each other for receiving thereturned signal at times which differ in relation to any differences inthe distances between said receiving antennas and the transponder; d.first means for mixing the signal received at the first receivingantenna with the transmitted signal and for producing a first beatsignal Fb₁ having a frequency fb₁ ; e. second means for mixing thesignal received at the second receiving antenna with the transmittedsignal and for producing a second beat signal Fb₂ having a frequency offb₂ ; f. means for measuring phase shifts φ₀ and φ'₀ including:
 1. afirst circuit coupled to the means for producing the first beat signalFb₁, said first circuit effecting production of a signal B₁ beginningwith the first beat of Fb₁ following the transmission of a frequency F₁at a time t₁ and ending upon completion of a predetermined number ofbeats of Fb₁ ;2. a second circuit coupled to the means for producing thesecond beat signal Fb₂, said second circuit effecting production of asignal B₂ beginning with the first beat of Fb₂ following thetransmission of the frequency F₁ at time t₁ and ending upon completionof a predetermined number of beats of Fb₂ ;
 3. means coupled to thefirst and second circuits for detecting the phase shift φ₀ between thebeginnings of the signals B₁, B₂ and for determining the phase shift φ'₀between the endings of the signals B₁, B₂ ; and g. computing meanscoupled to the phase shift measuring means and adapted for:3.determining an overall phase variation Δφ by performing the calculationΔφ=φ'₀ -φ₀ ;
 2. determining an approximate value φ.sub.Δφ for the phaseshift between the signals Fb₁ and Fb₂ at the time t₁ by performing thecalculation ##EQU24##
 3. determining the value of a constant k to thenearest integer by performing the appropriate one of the followingcalculations:if the sign of Δφ is positive, ##EQU25## if the sign of Δφis negative, ##EQU26##
 4. determining the accurate value of the phaseshift φ between the signals Fb₁ and Fb₂ by performing the appropriateone of the following calculations:if the sign of Δφ is positive, φ=φ₀+2kπ if the sign of Δφ is negative, φ=φ₀ -2kπ
 5. determining the azimuthα of the transponder by performing the calculation ##EQU27##
 4. Anapparatus as in claim 3 where the transmitting means comprises means forvarying the frequency of the transmitted signal in response to arepetitive sawtooth waveform.
 5. An apparatus as in claim 4 where saidtransmitting means includes a control loop for varying the period T ofthe sawtooth as a function of the distance between the radar system andthe transponder, said variation being effected to maintain apredetermined one of the beat frequencies substantially constant.
 6. Anapparatus as in claim 4 or 5 where the transmitting means comprisesmeans for producing the frequency F₁ at the beginning of each sawtooth.7. An apparatus as in claim 3, 4 or 5 where said first and secondcircuits of the means for measuring the phase shifts φ₀ and φ₀ 'comprise:a. first and second shaping circuits for converting the beatsignals Fb₁ and Fb₂ to representative square wave signals A₁ and A₂,respectively; b. first and second synchronizing circuits having inputscoupled to the respective shaping circuits for receiving the signals A₁and A₂, each synchronizing circuit further having a start input, a stopinput and an output, and producing a respective output signal B₁, B₂beginning with the first respective square wave signal A₁, A₂ followingreceipt at its start input of a signal at time t₁ ; c. means forapplying respective stop signals to the stop inputs of the synchronizingcircuits after time t₂ to effect termination of the signals B₁, B₂ whenequal numbers of square wave pulses have been produced by the first andsecond shaping circuits; and further where said means coupled to saidfirst and second circuits comprises: d. an exclusive-OR-gate havingfirst and second inputs coupled to the outputs of the first and secondsynchronizing circuits for receiving the signals B₁ and B₂ and forsuccessively producing at its output pulses of duration φ₀ and φ₀ '; e.counting means coupled to the output of the exclusive-OR-gate formeasuring the durations φ₀ and φ₀ '; and f. a memory coupled to theoutput of the counting means for storing the values of φ₀ and φ₀ '. 8.An apparatus as in claim 7 where said means for producing the first andsecond stop signals comprise:a. first and second AND-gates havingrespective inputs for receiving the signals A₁, B₁ and A₂, B₂ ; b. firstand second counters coupled to the outputs of the first and secondAND-gates, respectively, for counting the numbers of pulses N₁, N₂produced thereby; c. a comparator coupled to the outputs of the firstand second counters for producing a first signal when N₁ is greater thanor equal to N₂ and for producing a second signal when N₂ is greater thanor equal to N₁ ; and d. first and second AND-gates, each having a firstinput coupled to the comparator for receiving a respective one of thetwo signals produced thereby and having a second input for receiving asignal at time t₂, and having respective outputs coupled to the stopinputs of the first and second sychronizing circuits for applying therespective stop signals.